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    Poincare's Lemma and its Converse

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    For  phi E C2[R3 ! R3], curl grad phi = 0. Prove this. The converse is "Poincare's Lemma": if f E C1[R3 --> R3] and if curl f = 0, then f is a gradient, i.e., f = grad  for some  2 C2. Try it this way: if f = grad phi, then

    phi (x1, x2, x3) = phi(0)+ ....

    See why? Be that as it may, this function phi is perfectly well - defined. So start from scratch with this phi taken out of the air and see what it's gradient is, assuming curl f = 0.

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    For . If , then f is a gradient, ie., for some .

    Proof. (1) We will prove " If then "
    Let . ...

    Solution Summary

    A gradient is found using Poincare's Lemma. The solution is detailed and well presented.